c Color-Kinematics Duality for QCD Amplitudes · 2020. 11. 19. · 3 Henrik Johansson Uppsala U. &...

Henrik Johansson Uppsala U. & Nordita

Nov. 26, 2015

DESY Zeuthen

work with Alexander Ochirov arXiv: 1407.4772, 1507.00332

Color-Kinematics Duality for QCD Amplitudes

where we allow for arbitrary spins. All momenta are outgoing and the notation used is

ki,j,...,l = ki + kj + · · · + kl ,

si,j,...,l = (ki + kj + · · ·+ kl)2 ,

sij = (ki + kj)2 .

(3.7)

It is easy to see that the seven color factors (3.5) obey three commutation relations:

c1 − c2 = −c7 , c3 − c4 = −c7 , c5 − c6 = −c7 . (3.8)

Thus the color space of Atree6,3 is four-dimensional, which is consistent with the length of the

Melia basis (3.3). The primitive amplitudes therein can also be diagrammatically expanded

according to the color-ordered Feynman rules (A.1)

A123456 =n2

D2+

n4

D4+

n6

D6+

n7

D7, A123564 = −

n3

D3− n4

D4,

A125634 =n1

D1+

n3

D3+

n5

D5− n7

D7, A125346 = −

n5

D5− n6

D6,

(3.9)

where for brevity we have written the amplitude arguments as subscripts.

Now we wish to color-decompose the amplitude (3.4) such that we get exactly the Melia

basis primitives as kinematic coefficients. We observe that we can massage it into a com-

bination of the expressions in eq. (3.9) by using the commutation relations (3.8). Indeed,

if we eliminate the color factors c3, c6 and c7, we land on the following decomposition:

Atree6,3 =

c1n1

D1+

c2n2

D2+

(c1 − c2 + c4)n3

D3+

c4n4

D4+

c5n5

D5+

(c2 − c1 + c5)n6

D6+

(c2 − c1)n7

D7

= c2A123456 + c1 A125634 + (c2 − c4)A123564 + (c1 − c5)A125346

≡ C123456 A123456 + C125634 A125634 + C123564 A123564 + C125346 A125346 .(3.10)

The precise expressions for C12...’s, the color coefficients of the primitives, are subject to

the commutation identities (3.8). Our choice here is the one that will be generalized in

section 3.3. Some traits of the general pattern can be seen from the following rendition of

the color coefficients:

C123456 =2 1

3 64 5, C123564 =

2 1

6

3 4

5

+

2 1

6

3 4

5

,

C125634 =2 1

5 46 3, C125346 =

2 1

4

5 6

3

+

2 1

4

5 6

3

,

(3.11)

where we choose to draw some of the color diagrams in a nonplanar fashion in order to

preserve the cyclic ordering dictated by the primitive amplitudes. In other words, although

the color-ordered primitives (3.9) are composed only of planar Feynman diagrams, their

color coefficients (3.11) contain both planar and nonplanar color diagrams. More than that,

we observe that this non-planarity is related to the bracket “nestedness” for configurations{

3{5 6}4}

and{

5{3 4}6}

, as will be increasingly clear for higher-multiplicity examples.

– 12 –

!   Motivation & review: color-kinematics duality !  Various gravity/gauge theories

!   Generalization to QCD tree amplitudes

!   New color decomposition

!   Primitive amplitude relations for QCD

!   Simple one loop application !  One loop 4pt amplitude

!   Conclusion

Outline

Color-kinematics duality

3

A(L)n

Color-kinematics duality for pure YM YM theories are controlled by a hidden kinematic Lie algebra

• Amplitude expanded in terms of cubic graphs:

Color & kinematic numerators satisfy same relations:

Jacobi identity

propagators

color factors

kinematic numerators

Bern, Carrasco, HJ

Generalized gauge transformations

5

In general Feynman diagrams do not obey C-K duality

•  Four-gluon vertex absorbed into cubic graphs à ambiguity •  Feynman diagrams are gauge-dependent

à no reason to expect C-K duality to be present in all gauges

Amplitudes are invariant under “generalized gauge transformations”

Claim: starting from a general gauge there exists transformations that makes the numerators obey the duality !

but not duality:

such that

?

Bern, Carrasco, HJ (’08) (‘10)

Gauge-invariant relations

BCJ rels. proven via string theory by Bjerrum-Bohr, Damgaard, Vanhove; Stieberger (’09)

and field theory proofs through BCFW: Feng, Huang, Jia; Chen, Du, Feng (’10 -’11)

Relations used in string calcs: Mafra, Stieberger, Schlotterer (’11 -’13)

Relations used by Cachazo, He, Yuan to motivate CHY and scattering eqns (’13)

BCJ relations (‘08) (n-3)! basis

n�1X

i=1

A(1, 2, . . . , i, n, i+ 1, . . . , n� 1) = 0 U(1) decoupling

Kleiss-Kuijf relations (‘89)

(n-2)! basis

A(1, 2, . . . , n� 1, n) = A(n, 1, 2, . . . , n� 1) Cyclicity à (n-1)! basis

Gravity is a double copy of YM

!   The two numerators can differ by a generalized gauge transformation à only one copy needs to satisfy the kinematic algebra

!   The two numerators can differ by the external/internal states à graviton, dilaton, axion (B-tensor), matter amplitudes

!   The two numerators can belong to different theories à give a host of different gravitational theories

Gravity amplitudes obtained by replacing color with kinematics

double copy Bern, Carrasco, HJ

Squaring of YM theory Gravity processes = squares of gauge theory ones - entire S-matrix

Gravity Yang-Mills

pure Yang-Mills → Einstein gravity + dilaton + axion

N =4 super-YM → N =8 supergravity

→�

→�

E.g.

squared numerators

Bern, Carrasco, HJ (’10)

Which “gauge” theories obey C-K duality

!   Pure N=0,1,2,4 super-Yang-Mills (any dimension)

!   Self-dual Yang-Mills theory O’Connell, Monteiro (’11) !   Heterotic string theory Stieberger, Taylor (’14)

!   Yang-Mills + F 3 theory Broedel, Dixon (’12)

!   QCD, super-QCD, higher-dim QCD HJ, Ochirov (’15)

!   Generic matter coupled to N = 0,1,2,4 super-Yang-Mills

!   Spontaneously broken N = 0,2,4 SYM

!   Yang-Mills + scalar ϕ3 theory !   Bi-adjoint scalar ϕ3 theory !   NLSM/Chiral Lagrangian !  D=3 Bagger-Lambert-Gustavsson theory (Chern-Simons-matter)

Bern, Carrasco, HJ (’08) Bjerrum-Bohr, Damgaard, Vanhove; Stieberger; Feng et al. Mafra, Schlotterer, etc (’08-’11)

Chiodaroli, Gunaydin, Roiban; HJ, Ochirov (’14)

Chiodaroli, Gunaydin, HJ, Roiban (’14) Bern, de Freitas, Wong (’99), Bern, Dennen, Huang; Du, Feng, Fu; Bjerrum-Bohr, Damgaard, Monteiro, O’Connell

Bargheer, He, McLoughlin; Huang, HJ, Lee (’12 -’13)

Chen, Du (’13) �

Chiodaroli, Gunaydin, HJ, Roiban (’15)

Which gravity theories are double copies

!   Pure N=4,5,6,8 supergravity (2 < D < 11) Bern, Carrasco, HJ (’08 -’10)

!   Einstein gravity and pure N=1,2,3 supergravity HJ, Ochirov (’14)

!   Self-dual gravity O’Connell, Monteiro (’11)

!   Closed string theories Mafra, Schlotterer, Stieberger (’11); Stieberger, Taylor (’14)

!   Einstein + R 3 theory Broedel, Dixon (’12)

!   Abelian matter coupled to supergravity HJ, Ochirov (’14 - ’15)

!   SYM coupled to supergravity Chiodaroli, Gunaydin, HJ, Roiban (’14)

!   Spontaneously broken YM-Einstein gravity Chiodaroli, Gunaydin, HJ, Roiban (’15)

!   D=3 supergravity (BLG Chern-Simons-matter theory)2

!   Born-Infeld, DBI, Galileon theories Cachazo, He, Yuan (’14)

Bargheer, He, McLoughlin; Huang, HJ, Lee (’12 -’13)

Self-dual kinematic Lie algebra

Monteiro and O’Connell (’11) Self dual YM in light-cone gauge:

Diffeomorphism symmetry hidden in YM theory!

The X(p1, p2) are YM vertices of type ++-- helicity.

Self dual sector gives +++…+ amplitudes (S-matrix is one-loop exact)

Lie Algebra: YM vertex

Generators of diffeomorphism invariance:

Boels, Isermann, Monteiro, O'Connell

Color-Kinematics Duality for QCD

12

Defining QCD ‘QCD’ is taken to be the following theory:

SU(Nc) YM + Nf massive quarks

In fact, everything I say will also apply to:

Gc YM + Nf massive complex-rep. fermions

in D dimensions or SUSY extended SQCD

/scalars

Only use two Lie-algebra properties adjoint repr. or gluon, or vector multipl. Jacobi Id.

Commutation Id.

fund. repr. or fermion, or complex scalar, or matter multipl.

Duality:

Amplitude presentation for QCD

QCD amplitude with k quark lines of distinct flavor:

sum is over all cubic gluon--quark graphs with vertices Color factors are built out of ci fabc , T a

i|

k \ n 3 4 5 6 7 8

0 1 3 15 105 945 10395

1 1 3 15 105 945 10395

2 - 1 5 35 315 3465

3 - - - 7 63 693

4 - - - - - 99

ν(n, k) = (2n−5)!!(2k−1)!! for 2k ≤ n

Table 1: Number of cubic graphs, ν(n, k), in the full n-point amplitude with k distinguishablequark-antiquark pairs and (n− 2k) gluons.

and they carry no new physical information with respect to the cubic interactions. This

nontrivial statement is made apparent by the on-shell recursion [6, 7], which relies only on

input from the three-point amplitudes of the theory.

For the color structures the redundancy of quartic interactions is clear from inspecting

the four-gluon Feynman vertex

a,λ

b, µ c, ν

d, ρ

= ig2

2

[

fabef ecd(

gλνgµρ − gλρgµν)

+ f bcef eda(

gλνgµρ − gλµgνρ)

+facef ebd(

gλµgνρ − gλρgµν)]

,

(2.1)

where we, for later convenience, use imaginary structure constants fabc = i√2fabc. This

vertex contains the same color factors as the s-, t- and u-channel diagrams that are con-

structed from the three-gluon vertices. Hence the quartic vertex can always be absorbed in

some way into cubic trees without changing the general color structure of a QCD amplitude.

Therefore, without loss of generality we can write a QCD tree amplitude in terms of

cubic graphs only. This gives us an expansion of the n-point tree amplitude of the form2

Atreen,k = gn−2

ν(n,k)∑

cubic graphs Γi

cini

Di, (2.2)

where ci are color factors, ni are kinematic numerators, and Di are denominators encoding

the propagator structure of the cubic graphs. The denominators may contain masses,

corresponding to massive quark propagators.

Amplitudes with multiple quarks of the same flavor and mass can be obtained from

distinct-flavor amplitudes by setting masses to be equal and summing over permutations of

quarks. Therefore, we do not lose generality by taking all k quark-antiquark pairs to have

distinct flavor and mass. For explicitness, in table 1 we provide total counts of cubic graphs

for different amplitudes up to eight particles and four quark pairs. This count appears in

eq. (2.2) as ν(n, k). It agrees with the usual counting of standard QCD Feynman diagrams

restricted to those diagrams that only have trivalent vertices.

2Following ref. [31] we absorb all factors of i into the numerators, which is convenient for tree amplitudes.

The numerators in ref. [50] have a factor of −i pulled out relative to this convention.

– 4 –

Number of cubic tree-level graphs

A(L)n,k

A(L)m =

X

i

ZdLD`

(2⇡)LD

1

Si

niciDi

HJ, Ochirov

n=5 k=2 example Look at 3 Feynman diagrams out of 5 in total:

Non-trivial tree-level example

1−, i2+,

3−, k 4+, l

5, a =i√2

1s15s34

T aimT b

mTbkl 〈1|ε5|1+5|3〉 [24] = c1n1

D1

1−, i2+,

3−, k 4+, l

5, a = − i√2

1s25s34

T bimT a

mTbkl 〈13〉 [2|ε5|2+5|4] = c2n2

D2

1−, i2+,

3−, k 4+, l

5, a =i√2

1s12s34

fabcT biT

ckl

(

〈1| ε5|2] 〈3| 5|4] − 〈1| 5|2] 〈3| ε5|4]

− 2 〈13〉 [24]((k1+k2)·ε5))

=c5n5

D5

commutation relation:

c1 − c2 = −c5 ⇔ n1 − n2 = −n5

12 / 26

Non-trivial tree-level example

1−, i2+,

3−, k 4+, l

5, a =i√2

1s15s34

T aimT b

mTbkl 〈1|ε5|1+5|3〉 [24] = c1n1

D1

1−, i2+,

3−, k 4+, l

5, a = − i√2

1s25s34

T bimT a

mTbkl 〈13〉 [2|ε5|2+5|4] = c2n2

D2

1−, i2+,

3−, k 4+, l

5, a =i√2

1s12s34

fabcT biT

ckl

(

〈1| ε5|2] 〈3| 5|4] − 〈1| 5|2] 〈3| ε5|4]

− 2 〈13〉 [24]((k1+k2)·ε5))

=c5n5

D5

commutation relation:

c1 − c2 = −c5 ⇔ n1 − n2 = −n5

12 / 26

Not gauge invariant, but satisfy color-kinematics duality

Color decomposition

only gluons:

(a) − =

(b) − =

Figure 2: Color-algebra relations in the adjoint (a) and fundamental representation (b).The color-kinematics duality requires that the kinematic numerators satisfy the correspondingkinematic-algebra relations, which can be represented by the same graphs.

This leads to a basis of linearly-independent color structures. In the pure-gluon case it

gives the familiar color-trace decomposition [36, 57–59],

Atreen,0 =

∑

σ∈Sn−1({2,...,n})

Tr(

T a1T aσ(2) . . . T aσ(n))

A(1,σ(2), . . . ,σ(n)) , (2.8)

where the sum is over (n−1)! primitives because the cyclic symmetry of the trace allows one

to fix the first argument.3 In a similar but much more cumbersome way, the color content

of amplitudes with k quark-antiquark pairs and (n− 2k) gluons can be reduced [36, 60–67]

to color structures of the following type:

1

Npc

(

T a2k+1 . . . T al1)

i1α1

(

T al1+1 . . . T al2)

i2α2. . .

(

T alk−1+1 . . . T an)

ikαk, (2.9)

where lf ∈ {2k + 1, . . . , n}, αf ∈ {1, . . . , k}, and (p + 1) counts the number of disjoint

cycles in the permutation α.

An obvious feature of the above SU(Nc) decomposition is that it is specific to the

gauge group. A more interesting drawback is that it usually maps the color space to

a basis that is larger than the number of linearly independent color factors ci. Hence

the resulting kinematic coefficients – the color-ordered amplitudes – are not the minimal

set of primitives. In other words, they are not independent. Indeed, the purely gluonic

color-ordered amplitudes can be reduced using the Kleiss-Kuijf (KK) relations [30]. By

construction, the number of the primitives independent under such color-algebra relations

must coincide with the number of linearly independent color factors. The KK amplitude

relations can be written as

A(1,β, 2,α) = (−1)|β|∑

σ∈α""βT

A(1, 2,σ) , (2.10)

where the sum runs over the shuffle product of the ordered sets α and βT , the latter being

β in reverse order. This gives all partially ordered permutations that respect the element

order of the two sets. The KK relations let us fix the second argument in the pure gluon

primitives, and hence they reduce the basis to (n− 2)! elements – the KK basis.

3The reversal symmetry A(1, 2, . . . , n) = (−1)nA(n, . . . , 2, 1) further reduces that count to (n− 1)!/2.

– 6 –

(a) − =

(b) − =

Figure 2: Color-algebra relations in the adjoint (a) and fundamental representation (b).The color-kinematics duality requires that the kinematic numerators satisfy the correspondingkinematic-algebra relations, which can be represented by the same graphs.

This leads to a basis of linearly-independent color structures. In the pure-gluon case it

gives the familiar color-trace decomposition [36, 57–59],

Atreen,0 =

∑

σ∈Sn−1({2,...,n})

Tr(

T a1T aσ(2) . . . T aσ(n))

A(1,σ(2), . . . ,σ(n)) , (2.8)

where the sum is over (n−1)! primitives because the cyclic symmetry of the trace allows one

to fix the first argument.3 In a similar but much more cumbersome way, the color content

of amplitudes with k quark-antiquark pairs and (n− 2k) gluons can be reduced [36, 60–67]

to color structures of the following type:

1

Npc

(

T a2k+1 . . . T al1)

i1α1

(

T al1+1 . . . T al2)

i2α2. . .

(

T alk−1+1 . . . T an)

ikαk, (2.9)

where lf ∈ {2k + 1, . . . , n}, αf ∈ {1, . . . , k}, and (p + 1) counts the number of disjoint

cycles in the permutation α.

An obvious feature of the above SU(Nc) decomposition is that it is specific to the

gauge group. A more interesting drawback is that it usually maps the color space to

a basis that is larger than the number of linearly independent color factors ci. Hence

the resulting kinematic coefficients – the color-ordered amplitudes – are not the minimal

set of primitives. In other words, they are not independent. Indeed, the purely gluonic

color-ordered amplitudes can be reduced using the Kleiss-Kuijf (KK) relations [30]. By

construction, the number of the primitives independent under such color-algebra relations

must coincide with the number of linearly independent color factors. The KK amplitude

relations can be written as

A(1,β, 2,α) = (−1)|β|∑

σ∈α""βT

A(1, 2,σ) , (2.10)

where the sum runs over the shuffle product of the ordered sets α and βT , the latter being

β in reverse order. This gives all partially ordered permutations that respect the element

order of the two sets. The KK relations let us fix the second argument in the pure gluon

primitives, and hence they reduce the basis to (n− 2)! elements – the KK basis.

3The reversal symmetry A(1, 2, . . . , n) = (−1)nA(n, . . . , 2, 1) further reduces that count to (n− 1)!/2.

– 6 –

with quarks more complicated

2 1

σ(3) σ(4) . . . σ(n)

Figure 3: Multi-peripheral cubic diagram for the color factors in formulas (2.11) and (2.12). Allpermuted legs are gluons, while the horizontal line can be either a quark or a gluon line.

An interesting exception to the redundancy of the aforementioned decomposition al-

gorithm is the case of the amplitude with a single quark line [35, 36],

Atreen,1 =

∑

σ∈Sn−2({3,...,n})

(

T aσ(3) . . . T aσ(n))

2i1A(1, 2,σ(3), . . . ,σ(n)) , (2.11)

where we have denoted the quark-antiquark pair by a bar below and above the labels, i.e.

as 1 and 2. The sum is over the basis of (n− 2)! primitives, which are already independent

under color-algebra relations. Moreover, the color factors in eq. (2.11) are valid for any

gauge group, as can be guessed from the absence of explicit factors of Nc.

The reason for the nice properties of eq. (2.11) is that this example happens to coincide

with another basis of color factors that have these properties more generally. For k < 2 the

color factors in this basis correspond to the “multi-peripheral” graphs shown in figure 3.

The above k = 1 case can be easily mapped to the pure-gluon configuration by replacing

T a → T aadj. This gives the decomposition of Del Duca, Dixon and Maltoni (DDM) [33, 34]:

Atreen,0 =

∑

σ∈Sn−2({3,...,n})

f a2aσ(3)b1 f b1aσ(4)b2 . . . f bn−3aσ(n)a1 A(1, 2,σ(3), . . . ,σ(n)) . (2.12)

Note that it is a substantial improvement over the trace decomposition (2.8), since it avoids

using (n− 2)2(n− 3)! primitives altogether.

The details become more involved when considering generalizations along the lines of

the DDM decomposition to more than one quark line. A basis of amplitude primitives for

generic tree amplitudes in QCD, which have the same count as the number of indepen-

dent color structures, was recently found by Melia in refs. [37, 38]. However, it remained

unknown what are the corresponding color coefficients in a decomposition using that basis.

In section 3 we discuss the Melia basis in detail, and give the complete amplitude

color decomposition in terms of this basis. The new decomposition can be thought of as a

natural generalization of the DDM decomposition to the case of k quark-antiquark pairs,

in analogy to how the Melia basis is the multi-quark generalization of the KK basis for

amplitude primitives. Similarly to the multi-peripheral formulas (2.11) and (2.12), the color

coefficients of the new decomposition will be constructed from the cubic color factors ciand will thus be valid for any gauge group (and any group representation for the quarks).

2.2 Color-kinematics duality

Let us return to the trivalent graph expansion (2.2) of a gauge-theory amplitude, involving

kinematic numerator factors ni, color factors ci and denominatorsDi. As already explained,

the numerators are not uniquely defined, since, for instance, the quartic-vertex contact

– 7 –

e.g. k=1

⇠

2 1

σ(3) σ(4) . . . σ(n)




Atreen,1 =

∑

σ∈Sn−2({3,...,n})

(

T aσ(3) . . . T aσ(n))

2i1A(1, 2,σ(3), . . . ,σ(n)) , (2.11)










Atreen,0 =

∑

σ∈Sn−2({3,...,n})



















– 7 –

trace basis decomposition SU(Nc)

Del Duca, Dixon, Maltoni (DDM) basis

2 1

σ(3) σ(4) . . . σ(n)




Atreen,1 =

∑

σ∈Sn−2({3,...,n})

(

T aσ(3) . . . T aσ(n))

2i1A(1, 2,σ(3), . . . ,σ(n)) , (2.11)










Atreen,0 =

∑

σ∈Sn−2({3,...,n})



















– 7 –

Properties: valid for any G, gives small (n - 2)! basis

Mangano, …

Basis of planar (color-ordered) tree amplitudes:

Dyck words

is manifest in the adjoint representation, and can be imposed in the fundamental case (see

figure 1). Under the projection that removes crossed fermion line diagrams, the antisym-

metry of the vertices is maintained (since zero is antisymmetric) thus all the fermion-case

KK relations are inherited from the pure-gluon case unscathed, though some of them are

reduced to redundant equations, or even to trivial 0 = 0 equations. This implies that the

fermion-case basis of primitives has to be a subset of the original basis of (n−2)! primitives.

In fact, the (n − 2)! basis decomposes into k! = (n/2)! cases of inequivalent permutations

of the quarks (maintaining the antiquark ordering), out of which only one has no flavor

crossings. Hence, the basis surviving the projection is reduced in size by a factor 1/(n/2)!.

Indeed, the Melia basis for pure-fermion amplitudes has dimension (n − 2)!/(n/2)! [37],

and in the general mixed quark-gluon case the dimension is (n− 2)!/k! [38].

The fact that the basis of primitives were derived in refs. [37, 38] for the case of

adjoint particles and color-ordered planar amplitudes is not a problem. The same planar

amplitudes are also a basis of primitives in the mixed adjoint-fundamental case that we

encounter in QCD. We illustrate this through examples, and then give the general color

decomposition for QCD.

3.1 Pure-quark example: n = 6, k = 3

Let us now explain the details of the Melia basis using an instructive pure-quark example.

The basis of primitive amplitudes for k = n/2 quark-antiquark pairs is given by [37]5

{

A(1, 2,σ)∣∣ σ ∈ Dyckk−1

}

, (3.1)

where the last (n − 2) arguments must form a valid Dyck word out of the quarks and

antiquark labels.6 These words are defined as strings of X and Y letters, (k − 1) of each,

such that the number of X’s preceding each Y is greater than the number of preceding

Y’s. A more illuminating representation is obtained by realizing that Dyck words precisely

correspond to well-formed brackets, with X’s playing the role of opening brackets and Y’s

being closing brackets. For n = 6 there are two such words, XYXY and XXYY, equivalent

to the brackets {}{} and {{}}, respectively. An example of an invalid Dyck word is XYYX,

and it translates to the ill-formed brackets {}}{.Of course, the brackets themselves are not sufficient to specify the particle labels, in-

stead they only specify the particle type. The X’s or opening brackets can be identified

with quarks and Y’s or closing brackets with antiquarks (or, more precisely, the fundamen-

tal and anti-fundamental representation, respectively). To form the Dyck words relevant

for eq. (3.1), each valid bracket combination needs to be populated by particle labels. To

be specific, if we have quark flavor lines 3← 4 and 5← 6, then there are exactly two label

assignments per bracket combination that leave the flavor lines uncrossed:

XYXY ⇒ (3, 4, 5, 6), (5, 6, 3, 4) ⇔ {3 4}{5 6}, {5 6}{3 4} , (3.2a)

XXYY ⇒ (3, 5, 6, 4), (5, 3, 4, 6) ⇔{

3{5 6}4}

,{

5{3 4}6}

. (3.2b)

5The precise basis used in ref. [37] is slightly different:{

A(2, 1,σ)∣

∣ σ ∈ Dyckk−1

}

in our notation.6Recall that we mark quarks and antiquarks with underscores and overscores, respectively.

– 10 –

is manifest in the adjoint representation, and can be imposed in the fundamental case (see

figure 1). Under the projection that removes crossed fermion line diagrams, the antisym-

metry of the vertices is maintained (since zero is antisymmetric) thus all the fermion-case

KK relations are inherited from the pure-gluon case unscathed, though some of them are

reduced to redundant equations, or even to trivial 0 = 0 equations. This implies that the

fermion-case basis of primitives has to be a subset of the original basis of (n−2)! primitives.

In fact, the (n − 2)! basis decomposes into k! = (n/2)! cases of inequivalent permutations

of the quarks (maintaining the antiquark ordering), out of which only one has no flavor

crossings. Hence, the basis surviving the projection is reduced in size by a factor 1/(n/2)!.

Indeed, the Melia basis for pure-fermion amplitudes has dimension (n − 2)!/(n/2)! [37],

and in the general mixed quark-gluon case the dimension is (n− 2)!/k! [38].

The fact that the basis of primitives were derived in refs. [37, 38] for the case of

adjoint particles and color-ordered planar amplitudes is not a problem. The same planar

amplitudes are also a basis of primitives in the mixed adjoint-fundamental case that we

encounter in QCD. We illustrate this through examples, and then give the general color

decomposition for QCD.

3.1 Pure-quark example: n = 6, k = 3

Let us now explain the details of the Melia basis using an instructive pure-quark example.

The basis of primitive amplitudes for k = n/2 quark-antiquark pairs is given by [37]5

{

A(1, 2,σ)∣∣ σ ∈ Dyckk−1

}

, (3.1)

where the last (n − 2) arguments must form a valid Dyck word out of the quarks and

antiquark labels.6 These words are defined as strings of X and Y letters, (k − 1) of each,

such that the number of X’s preceding each Y is greater than the number of preceding

Y’s. A more illuminating representation is obtained by realizing that Dyck words precisely

correspond to well-formed brackets, with X’s playing the role of opening brackets and Y’s

being closing brackets. For n = 6 there are two such words, XYXY and XXYY, equivalent

to the brackets {}{} and {{}}, respectively. An example of an invalid Dyck word is XYYX,

and it translates to the ill-formed brackets {}}{.Of course, the brackets themselves are not sufficient to specify the particle labels, in-

stead they only specify the particle type. The X’s or opening brackets can be identified

with quarks and Y’s or closing brackets with antiquarks (or, more precisely, the fundamen-

tal and anti-fundamental representation, respectively). To form the Dyck words relevant

for eq. (3.1), each valid bracket combination needs to be populated by particle labels. To

be specific, if we have quark flavor lines 3← 4 and 5← 6, then there are exactly two label

assignments per bracket combination that leave the flavor lines uncrossed:

XYXY ⇒ (3, 4, 5, 6), (5, 6, 3, 4) ⇔ {3 4}{5 6}, {5 6}{3 4} , (3.2a)

XXYY ⇒ (3, 5, 6, 4), (5, 3, 4, 6) ⇔{

3{5 6}4}

,{

5{3 4}6}

. (3.2b)

5The precise basis used in ref. [37] is slightly different:{

A(2, 1,σ)∣

∣ σ ∈ Dyckk−1

}

in our notation.6Recall that we mark quarks and antiquarks with underscores and overscores, respectively.

– 10 –

Atree6,3 =

1 2

3

4 5

6 +1 2

5

6 3

4 +3 4

5

6 1

2 +3 4

1

2 5

6

+5 6

1

2 3

4 +5 6

3

4 1

2 +

12

3

4 5

6

Figure 4: Feynman diagrams for the six-quark amplitude Atree6,3 (1, 2, 3, 4, 5, 6).

These four valid Dyck words are written out using two different notations: the bar notation

used in eq. (2.11) and the bracket notation introduced above. As can be seen, the two

notations convey the same information, since each downstairs and upstairs bar can be

respectively identified with an opening and a closing bracket. In this paper we use both

notations interchangeably according to convenience. A very important aspect of either

notation is that each opening bracket corresponds to a unique closing bracket, and hence

each downstairs bar corresponds to a unique upstairs bar. These pairs of states are precisely

the quark-antiquark pairs of the same flavor. Thus the notation conveniently combines the

information about the gauge-group representation and flavor. In other words, no separate

notation is needed for specifying the flavor of the primitive amplitudes.

We have thus arrived at the Melia basis for n = 6, k = 3, which contains four primitives:

A(1, 2, 3, 4, 5, 6) , A(1, 2, 5, 6, 3, 4) , A(1, 2, 3, 5, 6, 4) and A(1, 2, 5, 3, 4, 6) . (3.3)

Now we will determine the color factors corresponding to the primitives and that are

appropriate for quarks in the fundamental representation. Recall that the full amplitude

can be written as

Atree6,3 =

c1n1

D1+

c2n2

D2+

c3n3

D3+

c4n4

D4+

c5n5

D5+

c6n6

D6+

c7n7

D7, (3.4)

where each term corresponds to a cubic Feynman diagram in figure 4. Their color factors

c1 = T ai1 T

bj ı2T

ai3 ı4T

bi5 ı6 , c2 = T b

i1Taj ı2T

ai3 ı4T

bi5 ı6 , (3.5a)

c3 = T ai3 T

bj ı4T

ai5 ı6T

bi1 ı2 , c4 = T b

i3Taj ı4T

ai5 ı6T

bi1 ı2 , (3.5b)

c5 = T ai5 T

bj ı6T

ai1 ı2T

bi3 ı4 , c6 = T b

i5Taj ı6T

ai1 ı2T

bi3 ı4 , (3.5c)

c7 = −fabcT ai1 ı2T

bi3 ı4T

ci5 ı6 , (3.5d)

can be read off according the rules in figure 1. For completeness and later use, we give the

kinematic content of three representative diagrams:

n1 = −i

4(u1γ

µ("k1,3,4+m1)γνv2)(u3γµv4)(u5γνv6) , D1 = (s1,3,4 −m2

1)s34s56 , (3.6a)

n2 = −i

4(u1γ

ν("k1,5,6+m1)γµv2)(u3γµv4)(u5γνv6) , D2 = (s1,5,6 −m2

1)s34s56 , (3.6b)

n7 = −i

4(u1γ

µv2)(u3γµv4)(u5("k1,2 − "k3,4)v6) + cyclic , D7 = s12s34s56 , (3.6c)

– 11 –

T. Melia only quarks

six-point example:

basis:


ki,j,...,l = ki + kj + · · · + kl ,

si,j,...,l = (ki + kj + · · ·+ kl)2 ,

sij = (ki + kj)2 .

(3.7)


c1 − c2 = −c7 , c3 − c4 = −c7 , c5 − c6 = −c7 . (3.8)




A123456 =n2

D2+

n4

D4+

n6

D6+

n7

D7, A123564 = −

n3

D3− n4

D4,

A125634 =n1

D1+

n3

D3+

n5

D5− n7

D7, A125346 = −

n5

D5− n6

D6,

(3.9)






Atree6,3 =

c1n1

D1+

c2n2

D2+

(c1 − c2 + c4)n3

D3+

c4n4

D4+

c5n5

D5+

(c2 − c1 + c5)n6

D6+

(c2 − c1)n7

D7

= c2A123456 + c1 A125634 + (c2 − c4)A123564 + (c1 − c5)A125346

≡ C123456 A123456 + C125634 A125634 + C123564 A123564 + C125346 A125346 .(3.10)





C123456 =2 1

3 64 5, C123564 =

2 1

6

3 4

5

+

2 1

6

3 4

5

,

C125634 =2 1

5 46 3, C125346 =

2 1

4

5 6

3

+

2 1

4

5 6

3

,

(3.11)






3{5 6}4}

and{

5{3 4}6}


– 12 –

Color coefficients:

HJ, Ochirov

Basis of planar (color-ordered) tree amplitudes:

Melia basis

T. Melia

gluons & quarks

size of basis:

Color decomposition, any any rep.

in terms of three primitives

A12534 = −n2

D2− n3

D3− n5

D5, A12345 = −

n1

D1− n4

D4+

n5

D5, A12354 =

n3

D3+

n4

D4. (3.17)

Note that they correspond to the single Dyck word XY, equivalent to the bracket {34},with the gluon label 5 inserted before, after and in the middle of the word, respectively.

Their color coefficients in the decomposition (3.16) are given by the following graphs:

C12534 =2 1

45 3

, C12345 =2 1

3 45

,

C12354 =

2 1

3 4

5

+

2 1

3 4

5

=

2 1

3 4

5

+

2 1

3 4

5

,

(3.18)

where C12354 is drawn both as (−c1 + c4) and (−c2 + c3) to emphasize that the nonplanar

diagrams cannot be removed by commutation relations. The pattern to take note of is that

the non-planarity occurs in the ordering {3 5 4} with the gluon sandwiched between the

quark brackets, which is reminiscent of the nested quark-antiquark pairs in section 3.1.

3.3 New color decomposition

In this section we formulate the new color decomposition for QCD.

We use the Melia basis for primitives with (n − 2k) gluons and k quark-lines [38]:

{

A(1, 2,σ)∣∣ σ ∈ Dyckk−1 × {gluon insertions}n−2k

}

. (3.19)

In the construction of this basis there are (2k−2)!/(k!(k−1)!) Dyck words prior to assigning

the particle labels inside the brackets. The quark labels can be assigned to (k− 1) slots in

(k − 1)! inequivalent ways. The antiquark labels have unique slot assignments after this,

since all quark lines have different flavors. Then the (n − 2k) gluons are assigned to any

place except between 1 and 2, which must stay adjacent. With each gluon inserted, the

number of available slots grows, starting from (2k − 1) up to (n − 2). Therefore, the size

of this color-algebra basis is

κ(n, k) =

empty brackets︷︸︸︷

(2k − 2)!

k!(k − 1)!×(k − 1)!

︸︷︷︸

dressed quark brackets

× (2k − 1)(2k) . . . (n− 2)︸︷︷︸

insertions of (n−2k) gluons

=(n− 2)!

k!, (3.20)

in agreement with the reasoning given in the beginning of section 3. See table 2 for the

explicit counts of the lower-multiplicity primitive amplitudes.

Now the new color decomposition for QCD is conveniently written as

Atreen,k =

κ(n,k)∑

σ∈Melia basis

C(1, 2,σ)A(1, 2,σ) , (3.21)

– 14 –


A12534 = −n2

D2− n3

D3− n5

D5, A12345 = −

n1

D1− n4

D4+

n5

D5, A12354 =

n3

D3+

n4

D4. (3.17)



C12534 =2 1

45 3

, C12345 =2 1

3 45

,

C12354 =

2 1

3 4

5

+

2 1

3 4

5

=

2 1

3 4

5

+

2 1

3 4

5

,

(3.18)








{


}

. (3.19)








κ(n, k) =


(2k − 2)!

k!(k − 1)!×(k − 1)!

︸︷︷︸


× (2k − 1)(2k) . . . (n− 2)︸︷︷︸


=(n− 2)!

k!, (3.20)




Atreen,k =

κ(n,k)∑

σ∈Melia basis

C(1, 2,σ)A(1, 2,σ) , (3.21)

– 14 –


A12534 = −n2

D2− n3

D3− n5

D5, A12345 = −

n1

D1− n4

D4+

n5

D5, A12354 =

n3

D3+

n4

D4. (3.17)



C12534 =2 1

45 3

, C12345 =2 1

3 45

,

C12354 =

2 1

3 4

5

+

2 1

3 4

5

=

2 1

3 4

5

+

2 1

3 4

5

,

(3.18)








{


}

. (3.19)








κ(n, k) =


(2k − 2)!

k!(k − 1)!×(k − 1)!

︸︷︷︸


× (2k − 1)(2k) . . . (n− 2)︸︷︷︸


=(n− 2)!

k!, (3.20)




Atreen,k =

κ(n,k)∑

σ∈Melia basis

C(1, 2,σ)A(1, 2,σ) , (3.21)

– 14 –

k \ n 3 4 5 6 7 8

0 1 2 6 24 120 720

1 1 2 6 24 120 720

2 - 1 3 12 60 360

3 - - - 4 20 120

4 - - - - - 30

Table 2: The number of n-point primitive amplitudes with k distinguishable quark pairs, inde-pendent under color-algebra relations, as given by the formula κ(n, k) = (n− 2)!/k!.

where A(1, 2,σ) are the usual color-ordered planar primitive amplitudes (defined by the

color-ordered Feynman rules in appendix A), and the color coefficients C(1, 2,σ) are non-

trivial objects that are the subject of the remaining discussion in this section.

Using the suggestive bracket notation, we can obtain the color coefficients using the

following replacement rules for the quark, antiquark and gluon labels:

C(1, 2,σ) = (−1)k−1 {2|σ|1}

∣∣∣∣∣

q → {q| T b⊗ Ξbl−1

q → |q}g → Ξ

ag

l

, (3.22)

where the integer l is the level of bracket “nestedness” for a given particle in the word

{2|σ|1}. In other words, l is the number of opening brackets minus the number of closing

brackets to the left of the particle. The bra {q| and the ket |q} now represent the funda-

mental and anti-fundamental color indices of a quark and an antiquark, or can equivalently

be understood as their the color wavefunctions. The object Ξal in eq. (3.22) is an operator

obtained by tensoring l copies of the Lie algebra,

Ξal =

l∑

s=1

1⊗ · · ·⊗ 1⊗s

︷︸︸︷

T a ⊗ 1⊗ · · · ⊗ 1⊗ 1︸︷︷︸

l

. (3.23)

The sum effectively runs over each slot s in the identity tensor product and inserts a genera-

tor with adjoint index a and in the appropriate representation, see figure 6. By convention,

each copy of the Lie algebra representation corresponds to a particular nestedness level,

starting from level l (the leftmost copy) and down to level one (the rightmost copy). Note

that the operators Ξal form a representation of the Lie algebra,

[

Ξal , Ξ

bl

]

= fabc Ξcl . (3.24)

This makes it natural to extend eq. (3.23) to the pure-gluon case by defining Ξa0 = T a

adj.

In the multi-sandwich formula (3.22) the quark and antiquark wavefunctions act only

on the Lie algebra copy at their corresponding nestedness level l. For example, {2| and |1}act on the level-one copy of the group representation, which is complex conjugated with

respect to the rest, as is seen in figure 6. This is indicated by the bar over the rightmost

unit operator in eq. (3.23) and the fact that in eq. (3.22) we use the bra {2| and the ket

|1} for the color wavefunctions of the antiquark 2 and quark 1, which is contrary to the

– 15 –

HJ, Ochirov

Gc, k,

Tensor representations

k \ n 3 4 5 6 7 8

0 1 2 6 24 120 720

1 1 2 6 24 120 720

2 - 1 3 12 60 360

3 - - - 4 20 120

4 - - - - - 30







C(1, 2,σ) = (−1)k−1 {2|σ|1}

∣∣∣∣∣

q → {q| T b⊗ Ξbl−1

q → |q}g → Ξ

ag

l

, (3.22)







Ξal =

l∑

s=1

1⊗ · · ·⊗ 1⊗s

︷︸︸︷

T a ⊗ 1⊗ · · · ⊗ 1⊗ 1︸︷︷︸

l

. (3.23)






[

Ξal , Ξ

bl

]

= fabc Ξcl . (3.24)


adj.






– 15 –

Tensor l copies of the gauge group Lie algebra:

Ξal =

a

...

l =

...

+

...

+

...

+ . . . +

...

Figure 6: Diagrammatic form of the operator Ξal . It is drawn as a single diagram with hollow

quark-gluon vertices, this represents summation over the possible locations where the gluon linecan attach.

convention for all other particles. This is due to the special role of the fermion line 1← 2.

This notational subtlety could in principle be avoided by complex conjugating that line into

1→ 2, i.e. by interchanging the roles of the quark and the antiquark.7 More generally, any

group representation copy can be complex conjugated, as it only changes the interpretation

of which particles are quarks and antiquarks. (In fact, any representation can be allowed

for each Lie algebra copy.)

The above formulas (3.22) and (3.23) unambiguously determine the color coefficients

in the color decomposition (3.21). For k = 1, 0 it is straightforward to see that our de-

composition coincides with the multi-peripheral formulas (2.11) and (2.12). Indeed, if all

permuted particles 3, . . . , n are gluons, they are replaced by Ξai1 = T

ai , producing pre-

cisely eq. (2.11). In case particles 1 and 2 are gluons as well, the nestedness level is zero,

and the adjoint-representation operators Ξai0 = T ai

adj yield the DDM decomposition (2.12).

For a more detailed example of using eq. (3.22), let us consider the six-point amplitude

discussed in section 3.1. One of the color coefficients in its decomposition is

C123456 = {2|{3|T a⊗ Ξa1|4}{5|T b⊗ Ξb

1|6}|1} = {2|{3|T a⊗ Ta|4}{5|T b⊗ T

b|6}|1}

= {2|T aTb|1}{3|T a|4}{5|T b|6} = (T bT a)i1ı2T

ai3ı4T

bi5ı6 ,

(3.25)

where we in the last step have translated it to a more conventional notation. A more

interesting example is the color coefficient

C123564 = {2|{3|T a⊗ Ξa1{5|T b⊗ Ξb

2|6}|4}|1}

= {2|{3|T a⊗ Ta{5|T b⊗ 1⊗ T

b|6}|4}|1} + {2|{3|T a⊗ Ta{5|T b⊗ T b⊗ 1 |6}|4}|1}

= {2|T aTb|1}{3|T a|4}{5|T b|6} + {2|T a|1}{3|T aT b|4}{5|T b|6} (3.26)

= (T bT a)i1ı2Tai3ı4T

bi5ı6 − T a

i1ı2(TaT b)i3ı4T

bi5ı6 .

The other two color coefficients for the six-point amplitude are, with less details, given by

C125634 = {2|{5|T a⊗ Ξa1|6}{3|T b⊗ Ξb

1|4}|1} = {2|T aTb|1}{3|T b|4}{5|T a|6} ,

C125346 = {2|{5|T a⊗ Ξa1{3|T b⊗ Ξb

2|4}|6}|1} = {2|T aTb|1}{3|T b|4}{5|T a|6}

+ {2|T a|1}{3|T b|4}{5|T aT b|6} .

(3.27)

7This would mean the Melia basis A(1, 2,σ), or, after relabeling, A(1,σ, n).

– 16 –

k \ n 3 4 5 6 7 8

0 1 2 6 24 120 720

1 1 2 6 24 120 720

2 - 1 3 12 60 360

3 - - - 4 20 120

4 - - - - - 30







C(1, 2,σ) = (−1)k−1 {2|σ|1}

∣∣∣∣∣

q → {q| T b⊗ Ξbl−1

q → |q}g → Ξ

ag

l

, (3.22)







Ξal =

l∑

s=1

1⊗ · · ·⊗ 1⊗s

︷︸︸︷

T a ⊗ 1⊗ · · · ⊗ 1⊗ 1︸︷︷︸

l

. (3.23)






[

Ξal , Ξ

bl

]

= fabc Ξcl . (3.24)


adj.






– 15 –

k \ n 3 4 5 6 7 8

0 1 2 6 24 120 720

1 1 2 6 24 120 720

2 - 1 3 12 60 360

3 - - - 4 20 120

4 - - - - - 30







C(1, 2,σ) = (−1)k−1 {2|σ|1}

∣∣∣∣∣

q → {q| T b⊗ Ξbl−1

q → |q}g → Ξ

ag

l

, (3.22)







Ξal =

l∑

s=1

1⊗ · · ·⊗ 1⊗s

︷︸︸︷

T a ⊗ 1⊗ · · · ⊗ 1⊗ 1︸︷︷︸

l

. (3.23)






[

Ξal , Ξ

bl

]

= fabc Ξcl . (3.24)


adj.






– 15 –

The are Lie algebra generators

k \ n 3 4 5 6 7 8

0 1 2 6 24 120 720

1 1 2 6 24 120 720

2 - 1 3 12 60 360

3 - - - 4 20 120

4 - - - - - 30







C(1, 2,σ) = (−1)k−1 {2|σ|1}

∣∣∣∣∣

q → {q| T b⊗ Ξbl−1

q → |q}g → Ξ

ag

l

, (3.22)







Ξal =

l∑

s=1

1⊗ · · ·⊗ 1⊗s

︷︸︸︷

T a ⊗ 1⊗ · · · ⊗ 1⊗ 1︸︷︷︸

l

. (3.23)






[

Ξal , Ξ

bl

]

= fabc Ξcl . (3.24)


adj.






– 15 –

Color coefficients

Color coefficients are given by ‘sandwich’ formulas:

k \ n 3 4 5 6 7 8

0 1 2 6 24 120 720

1 1 2 6 24 120 720

2 - 1 3 12 60 360

3 - - - 4 20 120

4 - - - - - 30







C(1, 2,σ) = (−1)k−1 {2|σ|1}

∣∣∣∣∣

q → {q| T b⊗ Ξbl−1

q → |q}g → Ξ

ag

l

, (3.22)







Ξal =

l∑

s=1

1⊗ · · ·⊗ 1⊗s

︷︸︸︷

T a ⊗ 1⊗ · · · ⊗ 1⊗ 1︸︷︷︸

l

. (3.23)






[

Ξal , Ξ

bl

]

= fabc Ξcl . (3.24)


adj.






– 15 –

color wave function

Ξal =

a

...

l =

...

+

...

+

...

+ . . . +

...

Figure 6: Diagrammatic form of the operator Ξal . It is drawn as a single diagram with hollow

quark-gluon vertices, this represents summation over the possible locations where the gluon linecan attach.

convention for all other particles. This is due to the special role of the fermion line 1← 2.

This notational subtlety could in principle be avoided by complex conjugating that line into

1→ 2, i.e. by interchanging the roles of the quark and the antiquark.7 More generally, any

group representation copy can be complex conjugated, as it only changes the interpretation

of which particles are quarks and antiquarks. (In fact, any representation can be allowed

for each Lie algebra copy.)

The above formulas (3.22) and (3.23) unambiguously determine the color coefficients

in the color decomposition (3.21). For k = 1, 0 it is straightforward to see that our de-

composition coincides with the multi-peripheral formulas (2.11) and (2.12). Indeed, if all

permuted particles 3, . . . , n are gluons, they are replaced by Ξai1 = T

ai , producing pre-

cisely eq. (2.11). In case particles 1 and 2 are gluons as well, the nestedness level is zero,

and the adjoint-representation operators Ξai0 = T ai

adj yield the DDM decomposition (2.12).

For a more detailed example of using eq. (3.22), let us consider the six-point amplitude

discussed in section 3.1. One of the color coefficients in its decomposition is

C123456 = {2|{3|T a⊗ Ξa1|4}{5|T b⊗ Ξb

1|6}|1} = {2|{3|T a⊗ Ta|4}{5|T b⊗ T

b|6}|1}

= {2|T aTb|1}{3|T a|4}{5|T b|6} = (T bT a)i1ı2T

ai3ı4T

bi5ı6 ,

(3.25)

where we in the last step have translated it to a more conventional notation. A more

interesting example is the color coefficient

C123564 = {2|{3|T a⊗ Ξa1{5|T b⊗ Ξb

2|6}|4}|1}

= {2|{3|T a⊗ Ta{5|T b⊗ 1⊗ T

b|6}|4}|1} + {2|{3|T a⊗ Ta{5|T b⊗ T b⊗ 1 |6}|4}|1}

= {2|T aTb|1}{3|T a|4}{5|T b|6} + {2|T a|1}{3|T aT b|4}{5|T b|6} (3.26)

= (T bT a)i1ı2Tai3ı4T

bi5ı6 − T a

i1ı2(TaT b)i3ı4T

bi5ı6 .

The other two color coefficients for the six-point amplitude are, with less details, given by

C125634 = {2|{5|T a⊗ Ξa1|6}{3|T b⊗ Ξb

1|4}|1} = {2|T aTb|1}{3|T b|4}{5|T a|6} ,

C125346 = {2|{5|T a⊗ Ξa1{3|T b⊗ Ξb

2|4}|6}|1} = {2|T aTb|1}{3|T b|4}{5|T a|6}

+ {2|T a|1}{3|T b|4}{5|T aT b|6} .

(3.27)

7This would mean the Melia basis A(1, 2,σ), or, after relabeling, A(1,σ, n).

– 16 –

For example, consider:


ki,j,...,l = ki + kj + · · · + kl ,

si,j,...,l = (ki + kj + · · ·+ kl)2 ,

sij = (ki + kj)2 .

(3.7)


c1 − c2 = −c7 , c3 − c4 = −c7 , c5 − c6 = −c7 . (3.8)




A123456 =n2

D2+

n4

D4+

n6

D6+

n7

D7, A123564 = −

n3

D3− n4

D4,

A125634 =n1

D1+

n3

D3+

n5

D5− n7

D7, A125346 = −

n5

D5− n6

D6,

(3.9)






Atree6,3 =

c1n1

D1+

c2n2

D2+

(c1 − c2 + c4)n3

D3+

c4n4

D4+

c5n5

D5+

(c2 − c1 + c5)n6

D6+

(c2 − c1)n7

D7

= c2A123456 + c1 A125634 + (c2 − c4)A123564 + (c1 − c5)A125346

≡ C123456 A123456 + C125634 A125634 + C123564 A123564 + C125346 A125346 .(3.10)





C123456 =2 1

3 64 5, C123564 =

2 1

6

3 4

5

+

2 1

6

3 4

5

,

C125634 =2 1

5 46 3, C125346 =

2 1

4

5 6

3

+

2 1

4

5 6

3

,

(3.11)






3{5 6}4}

and{

5{3 4}6}


– 12 –

HJ, Ochirov (proof by Melia)

Consider a high-multiplicity example:

Color coefficient diagrams

The four above color coefficients are indeed the ones given diagrammatically in eq. (3.11).

For the five-point amplitude discussed in section 3.2, our decomposition results in the

following color coefficients:

C12534 = − {2|Ξa51 {3|T b⊗ Ξb

1|4}|1} = − {2|T a5{3|T b⊗ Tb|4}|1} = − {2|T a5T

b|1}{3|T b|4} ,

C12345 = − {2|{3|T b⊗ Ξb1|4}Ξ

a51 |1} = − {2|{3|T b⊗ T

b|4}T a5 |1} = − {2|T bTa5 |1}{3|T b|4} ,

C12354 = − {2|{3|(T b⊗ Ξb1)Ξ

a52 |4}|1} (3.28)

= − {2|{3|(T b⊗ Tb)(1⊗ T

a5)|4}|1} − {2|{3|(T b⊗ Tb)(T a5⊗ 1 )|4}|1}

= − {2|T bTa5 |1}{3|T b|4} − {2|T b|1}{3|T bT a5 |4} ,

which coincide with the color diagrams in eq. (3.18).

While we do not provide a proof for the decomposition (3.21), we have explicitly

checked its validity for all quark-gluon configurations up to eight points, as well as the

nine-point amplitude with four quark lines and one gluon. To be more precise, the check

was done as follows. We expand the Melia basis of color-ordered primitives in kinematic

cubic diagrams,

A(1, 2,σ) =∑

σ-color-ordered cubic graphs Γi

± ni

Di, (3.29)

and solve for ni/Di in terms of the primitives A(1, 2,σ). When this solution is plugged back

into the color dressed cubic graph expansion (2.2), the color coefficients of the undetermined

ni can be shown to vanish under the color algebra (2.4). What is left is then an expansion

of the amplitude in terms of the primitives and some color factors ci. The color coefficients

C(1, 2,σ) of the primitives can then be read off from this expression. They are indeed given

exactly by eq. (3.22).

3.4 Higher-point example and color-coefficient diagram

Here we illustrate how the general formula (3.22) is applied to a high-multiplicity example.

Consider the color factor of the following 14-point primitive amplitude with six quark pairs

and two gluons (n = 14, k = 6):

A(1, 2, 13, 3, 5, 6, 4, 7, 9, 14, 11, 12, 10, 8) . (3.30)

We can use the cyclic property of planar amplitudes to move 1 to the end and then, to

make the nestedness of the primitive more apparent, replace the bar notation of the legs

with brackets:

{2 13{3{5 6}4}{7{9 14{11 12}10}8}1} . (3.31)

As before, the bar-bracket correspondence for the fermion line 1 ← 2 is opposite to all

other legs due to the different convention for Lie algebra representation on that line.

Using eq. (3.22) it is straightforward to obtain the expression for the color coefficient

of this primitive

C1,2,13,3,5,6,4,7,9,14,11,12,10,8 = − {2|Ξa131 {3|T b⊗ Ξb

1{5|T c⊗ Ξc2|6}|4} (3.32)

×{7|T d⊗ Ξd1{9|(T e⊗ Ξe

2)Ξa143 {11|T f⊗ Ξf

3 |12}|10}|8}|1} .

– 17 –




C12534 = − {2|Ξa51 {3|T b⊗ Ξb

1|4}|1} = − {2|T a5{3|T b⊗ Tb|4}|1} = − {2|T a5T

b|1}{3|T b|4} ,

C12345 = − {2|{3|T b⊗ Ξb1|4}Ξ

a51 |1} = − {2|{3|T b⊗ T

b|4}T a5 |1} = − {2|T bTa5 |1}{3|T b|4} ,

C12354 = − {2|{3|(T b⊗ Ξb1)Ξ

a52 |4}|1} (3.28)

= − {2|{3|(T b⊗ Tb)(1⊗ T

a5)|4}|1} − {2|{3|(T b⊗ Tb)(T a5⊗ 1 )|4}|1}

= − {2|T bTa5 |1}{3|T b|4} − {2|T b|1}{3|T bT a5 |4} ,






cubic diagrams,

A(1, 2,σ) =∑


± ni

Di, (3.29)











A(1, 2, 13, 3, 5, 6, 4, 7, 9, 14, 11, 12, 10, 8) . (3.30)



with brackets:

{2 13{3{5 6}4}{7{9 14{11 12}10}8}1} . (3.31)




of this primitive

C1,2,13,3,5,6,4,7,9,14,11,12,10,8 = − {2|Ξa131 {3|T b⊗ Ξb

1{5|T c⊗ Ξc2|6}|4} (3.32)

×{7|T d⊗ Ξd1{9|(T e⊗ Ξe

2)Ξa143 {11|T f⊗ Ξf

3 |12}|10}|8}|1} .

– 17 –

bra-(c)-ket structure

12

14 11 12

5 6 9 10

13 3 4 7 8

Figure 7: Diagrammatic representation for the color coefficient of the planar amplitudeA(1, 2, 13, 3, 5, 6, 4, 7, 9, 14, 11, 12, 10, 8), obtained by using the notation of figure 6. Note that thediagram has the same structure as the word {2 13{3{5 6}4}{7{9 14{11 12}10}8}1}.

While this formula is compact, it becomes a rather formidable expression if written

in terms of standard color factors. In order to understand it better, let us sketch out the

various standard color structures contained in it. For example, the first Ξa131 in eq. (3.32)

will give rise to the structure

{2|Ξa131 . . . |1} =

(

Ta13 . . .

)

ı2i1, (3.33)

whereas the second appearance of Ξ gives

{2| . . . {3|T a⊗ Ξa1 . . . |4} . . . |1} =

(

T a . . .)

i3 ı4

(

. . . Ta. . .

)

ı2i1. (3.34)

For l ≥ 2 the operator Ξal consists of a sum of tensor products, thus nested curly brackets

imply summation over different possibilities of inserting the generator T a. For example,

the third Ξ in eq. (3.32) gives rise to two structures

{2| . . . {3 . . . {5|T b3⊗ Ξb

2|6}|4} . . . |1} =(

T b)

i5 ı6

(

. . . T b)

i3 ı4

(

. . . . . .)

ı2i1

+(

T b)

i5 ı6

(

. . .)

i3 ı4

(

. . . Tb. . .

)

ı2i1,

(3.35)

while the sixth occurrence of Ξ gives three contributions

{2| . . . {7 . . . {9| . . .Ξa143 . . . |10}|8}|1} =

(

. . . T a14 . . .)

i9 ı10

(

. . .)

i7 ı8

(

. . .)

ı2i1

+(

. . . . . .)

i9 ı10

(

. . . T a14)

i7 ı8

(

. . .)

ı2i1

+(

. . . . . .)

i9 ı10

(

. . .)

i7 ı8

(

. . . Ta14)

ı2i1.

(3.36)

Indeed, each Ξal gives rise to exactly l structures, which then are multiplied together. This

implies that the number of standard color factors hiding in eq. (3.32) can be counted by

multiplying the subscripts of the Ξ’s in this expression, giving

1× 1× 2× 1× 2× 3× 3 = 36 terms. (3.37)

Finally, we note that probably the best way to understand the color coefficient (3.32)

is to draw a diagram for it. Indeed, figure 7 contains the same information as the for-

mula (3.32). In particular, compare this diagram with the word given in eq. (3.31). The

diagram figure 7 is similar to a usual color factor diagram that describes the contractions

– 18 –




C12534 = − {2|Ξa51 {3|T b⊗ Ξb

1|4}|1} = − {2|T a5{3|T b⊗ Tb|4}|1} = − {2|T a5T

b|1}{3|T b|4} ,

C12345 = − {2|{3|T b⊗ Ξb1|4}Ξ

a51 |1} = − {2|{3|T b⊗ T

b|4}T a5 |1} = − {2|T bTa5 |1}{3|T b|4} ,

C12354 = − {2|{3|(T b⊗ Ξb1)Ξ

a52 |4}|1} (3.28)

= − {2|{3|(T b⊗ Tb)(1⊗ T

a5)|4}|1} − {2|{3|(T b⊗ Tb)(T a5⊗ 1 )|4}|1}

= − {2|T bTa5 |1}{3|T b|4} − {2|T b|1}{3|T bT a5 |4} ,






cubic diagrams,

A(1, 2,σ) =∑


± ni

Di, (3.29)











A(1, 2, 13, 3, 5, 6, 4, 7, 9, 14, 11, 12, 10, 8) . (3.30)



with brackets:

{2 13{3{5 6}4}{7{9 14{11 12}10}8}1} . (3.31)




of this primitive

C1,2,13,3,5,6,4,7,9,14,11,12,10,8 = − {2|Ξa131 {3|T b⊗ Ξb

1{5|T c⊗ Ξc2|6}|4} (3.32)

×{7|T d⊗ Ξd1{9|(T e⊗ Ξe

2)Ξa143 {11|T f⊗ Ξf

3 |12}|10}|8}|1} .

– 17 –

⇥2 2 3 3 = 36 ⇥ ⇥

Amplitude relations: example

commutation rel. holds:

4.1 Four-point example: n = 4, k = 1

One of the simplest massive quark amplitudes involves one massive quark-antiquark pair

and two gluons. It has three Feynman diagrams:

1, i 2,

3, a 4, b

= − i

2

T aikT bk

s13−m2(u1 "ε3("k1,3+m)"ε4v2) =

c1n1

D1, (4.1a)

1, i 2,

4, b 3, a

= − i

2

T bikT ak

s14−m2(u1 "ε4("k1,4+m)"ε3v2) =

c2n2

D2, (4.1b)

1, i 2,

3, a 4, b

=i

2

fabcT ci

s12

(

2(k4 ·ε3)(u1 "ε4v2)− 2(k3 ·ε4)(u1 "ε3v2)

+ (ε3 ·ε4)(u1("k3 − "k4)v2))

=c3n3

D3,

(4.1c)

where the spins are left unspecified. These diagrams correspond precisely to the ones in

the fundamental commutation relation (2.4b), which in terms of color factors read

c1 − c2 = c3 . (4.2)

By the color-kinematics duality, the kinematic relation should then read

n1 − n2 = n3 . (4.3)

Indeed, the kinematic numerators of the above Feynman diagrams,

n1 = −i

2u1 "ε3("k1,3+m)"ε4v2 , n2 = −

i

2u1 "ε4("k1,4+m)"ε3v2 ,

n3 =i

2

(

2(k4 ·ε3)(u1 "ε4v2)− 2(k3 ·ε4)(u1 "ε3v2) + (ε3 ·ε4)(u1("k3 − "k4)v2))

,(4.4)

do satisfy eq. (4.3). To check this in detail one has to repeatedly use the Clifford algebra,

γµγν + γνγµ = 2ηµν , (4.5)

the Dirac equations and gluon transversality conditions,

u1("k1−m) = 0 , ("k2+m)v2 = 0 , ki · εi = 0 , (4.6)

as well as the mass-shell conditions for the quarks k21 = k22 = m2 and the gluons k23 = k24 = 0.

After several algebraic steps, one arrives at

n1 − n2 − n3 ∝ u1 "k1 "ε3 "ε4v2 + u1 "ε3 "ε4 "k2v2 − (ε3 ·ε4)(u1("k1 + "k2)v2) = 0 . (4.7)

Note that the numerators (4.4) are gauge-dependent through the polarization vectors

of the gluons, but the combination n1−n2−n3 is gauge-invariant (and zero). For a generic

amplitude the numerators of cubic graphs need to absorb the four-gluon interactions, which

– 20 –




1, i 2,

3, a 4, b

= − i

2

T aikT bk

s13−m2(u1 "ε3("k1,3+m)"ε4v2) =

c1n1

D1, (4.1a)

1, i 2,

4, b 3, a

= − i

2

T bikT ak

s14−m2(u1 "ε4("k1,4+m)"ε3v2) =

c2n2

D2, (4.1b)

1, i 2,

3, a 4, b

=i

2

fabcT ci

s12

(

2(k4 ·ε3)(u1 "ε4v2)− 2(k3 ·ε4)(u1 "ε3v2)

+ (ε3 ·ε4)(u1("k3 − "k4)v2))

=c3n3

D3,

(4.1c)



c1 − c2 = c3 . (4.2)


n1 − n2 = n3 . (4.3)


n1 = −i

2u1 "ε3("k1,3+m)"ε4v2 , n2 = −

i

2u1 "ε4("k1,4+m)"ε3v2 ,

n3 =i

2

(


,(4.4)




u1("k1−m) = 0 , ("k2+m)v2 = 0 , ki · εi = 0 , (4.6)







– 20 –




1, i 2,

3, a 4, b

= − i

2

T aikT bk

s13−m2(u1 "ε3("k1,3+m)"ε4v2) =

c1n1

D1, (4.1a)

1, i 2,

4, b 3, a

= − i

2

T bikT ak

s14−m2(u1 "ε4("k1,4+m)"ε3v2) =

c2n2

D2, (4.1b)

1, i 2,

3, a 4, b

=i

2

fabcT ci

s12

(

2(k4 ·ε3)(u1 "ε4v2)− 2(k3 ·ε4)(u1 "ε3v2)

+ (ε3 ·ε4)(u1("k3 − "k4)v2))

=c3n3

D3,

(4.1c)



c1 − c2 = c3 . (4.2)


n1 − n2 = n3 . (4.3)


n1 = −i

2u1 "ε3("k1,3+m)"ε4v2 , n2 = −

i

2u1 "ε4("k1,4+m)"ε3v2 ,

n3 =i

2

(


,(4.4)




u1("k1−m) = 0 , ("k2+m)v2 = 0 , ki · εi = 0 , (4.6)







– 20 –




1, i 2,

3, a 4, b

= − i

2

T aikT bk

s13−m2(u1 "ε3("k1,3+m)"ε4v2) =

c1n1

D1, (4.1a)

1, i 2,

4, b 3, a

= − i

2

T bikT ak

s14−m2(u1 "ε4("k1,4+m)"ε3v2) =

c2n2

D2, (4.1b)

1, i 2,

3, a 4, b

=i

2

fabcT ci

s12

(

2(k4 ·ε3)(u1 "ε4v2)− 2(k3 ·ε4)(u1 "ε3v2)

+ (ε3 ·ε4)(u1("k3 − "k4)v2))

=c3n3

D3,

(4.1c)



c1 − c2 = c3 . (4.2)


n1 − n2 = n3 . (4.3)


n1 = −i

2u1 "ε3("k1,3+m)"ε4v2 , n2 = −

i

2u1 "ε4("k1,4+m)"ε3v2 ,

n3 =i

2

(


,(4.4)




u1("k1−m) = 0 , ("k2+m)v2 = 0 , ki · εi = 0 , (4.6)







– 20 –




1, i 2,

3, a 4, b

= − i

2

T aikT bk

s13−m2(u1 "ε3("k1,3+m)"ε4v2) =

c1n1

D1, (4.1a)

1, i 2,

4, b 3, a

= − i

2

T bikT ak

s14−m2(u1 "ε4("k1,4+m)"ε3v2) =

c2n2

D2, (4.1b)

1, i 2,

3, a 4, b

=i

2

fabcT ci

s12

(

2(k4 ·ε3)(u1 "ε4v2)− 2(k3 ·ε4)(u1 "ε3v2)

+ (ε3 ·ε4)(u1("k3 − "k4)v2))

=c3n3

D3,

(4.1c)



c1 − c2 = c3 . (4.2)


n1 − n2 = n3 . (4.3)


n1 = −i

2u1 "ε3("k1,3+m)"ε4v2 , n2 = −

i

2u1 "ε4("k1,4+m)"ε3v2 ,

n3 =i

2

(


,(4.4)




u1("k1−m) = 0 , ("k2+m)v2 = 0 , ki · εi = 0 , (4.6)







– 20 –




1, i 2,

3, a 4, b

= − i

2

T aikT bk

s13−m2(u1 "ε3("k1,3+m)"ε4v2) =

c1n1

D1, (4.1a)

1, i 2,

4, b 3, a

= − i

2

T bikT ak

s14−m2(u1 "ε4("k1,4+m)"ε3v2) =

c2n2

D2, (4.1b)

1, i 2,

3, a 4, b

=i

2

fabcT ci

s12

(

2(k4 ·ε3)(u1 "ε4v2)− 2(k3 ·ε4)(u1 "ε3v2)

+ (ε3 ·ε4)(u1("k3 − "k4)v2))

=c3n3

D3,

(4.1c)



c1 − c2 = c3 . (4.2)


n1 − n2 = n3 . (4.3)


n1 = −i

2u1 "ε3("k1,3+m)"ε4v2 , n2 = −

i

2u1 "ε4("k1,4+m)"ε3v2 ,

n3 =i

2

(


,(4.4)




u1("k1−m) = 0 , ("k2+m)v2 = 0 , ki · εi = 0 , (4.6)







– 20 –




1, i 2,

3, a 4, b

= − i

2

T aikT bk

s13−m2(u1 "ε3("k1,3+m)"ε4v2) =

c1n1

D1, (4.1a)

1, i 2,

4, b 3, a

= − i

2

T bikT ak

s14−m2(u1 "ε4("k1,4+m)"ε3v2) =

c2n2

D2, (4.1b)

1, i 2,

3, a 4, b

=i

2

fabcT ci

s12

(

2(k4 ·ε3)(u1 "ε4v2)− 2(k3 ·ε4)(u1 "ε3v2)

+ (ε3 ·ε4)(u1("k3 − "k4)v2))

=c3n3

D3,

(4.1c)



c1 − c2 = c3 . (4.2)


n1 − n2 = n3 . (4.3)


n1 = −i

2u1 "ε3("k1,3+m)"ε4v2 , n2 = −

i

2u1 "ε4("k1,4+m)"ε3v2 ,

n3 =i

2

(


,(4.4)




u1("k1−m) = 0 , ("k2+m)v2 = 0 , ki · εi = 0 , (4.6)







– 20 –

invariably leads to ambiguities in defining the numerators, and thus the duality is not

manifest in general. However, in the above amplitude there is no quartic vertex, hence it

is not surprising that the duality holds automatically from the Feynman rules.

Having shown that the color-kinematics duality is present in this QCD amplitude, we

proceed using only the formal properties of the numerators. Expressing the color-dressed

amplitude through two color-ordered primitives is straightforward,

Atree4,1 =

3∑

i=1

cini

Di=

{

c1

(n1

D1+

n3

D3

)

+ c2

(n2

D2− n3

D3

)}

≡ c2A1234 + c1A1243 , (4.8)

where the color factors are of the multi-peripheral type, in accord with the decomposi-

tion (2.11). Using the duality, n3 = n1 − n2, we obtain a system of two equations,

A1234 =

(1

D2+

1

D3

)

n2 −n1

D3, A1243 =

(1

D1+

1

D3

)

n1 −n2

D3, (4.9)

which, after Gaussian elimination of n1, yields

A1234 =

(1

D2+

1

D3− D1

(D1+D3)D3

)

n2 −D1

D1+D3A1243 . (4.10)

The coefficient of n2 can be shown to be proportional to a vanishing sum of denominators

D1 +D2 +D3 = (s13 −m2) + (s14 −m2) + s12 = 0 , (4.11)

resulting in the following relation among the primitive amplitudes:

(s14 −m2)A1234 = (s13 −m2)A1243 . (4.12)

This is a straightforward generalization8 of the corresponding massless BCJ relation (2.15).

It allows us to express the full amplitude in terms of a single primitive,

Atree4,1 =

(

T a3ı2j

T a4 i1

+ T a4ı2j

T a3 i1

s14 −m2

s13 −m2

)

A1234 . (4.13)

4.2 Kinematic algebra for n = 5, k = 2

We proceed by showing that the color-kinematics duality is present in QCD amplitudes

with two quark lines of different flavors and masses. Let us return to the n = 5, k = 2

amplitude considered in section 3.2. Since the four-gluon vertex is also absent in this

amplitude, the numerators in eq. (3.12) are directly given by the Feynman rules, and we

expect the duality to hold automatically,

c1 − c2 = −c5 ⇔ n1 − n2 = −n5 , (4.14a)

c3 − c4 = c5 ⇔ n3 − n4 = n5 . (4.14b)

8Eq. (4.12) can be presented [54] as (k1 · k4)A1234 = (k1 · k3)A1243, making it formally identical to the

massless BCJ relation, but here we prefer to use momentum invariants with explicit mass dependence,

similar to Feynman propagators.

– 21 –







Atree4,1 =

3∑

i=1

cini

Di=

{

c1

(n1

D1+

n3

D3

)

+ c2

(n2

D2− n3

D3

)}

≡ c2A1234 + c1A1243 , (4.8)



A1234 =

(1

D2+

1

D3

)

n2 −n1

D3, A1243 =

(1

D1+

1

D3

)

n1 −n2

D3, (4.9)


A1234 =

(1

D2+

1

D3− D1

(D1+D3)D3

)

n2 −D1

D1+D3A1243 . (4.10)


D1 +D2 +D3 = (s13 −m2) + (s14 −m2) + s12 = 0 , (4.11)


(s14 −m2)A1234 = (s13 −m2)A1243 . (4.12)



Atree4,1 =

(

T a3ı2j

T a4 i1

+ T a4ı2j

T a3 i1

s14 −m2

s13 −m2

)

A1234 . (4.13)







c1 − c2 = −c5 ⇔ n1 − n2 = −n5 , (4.14a)

c3 − c4 = c5 ⇔ n3 − n4 = n5 . (4.14b)




– 21 –

Gaussian elimination of n1

= 0







Atree4,1 =

3∑

i=1

cini

Di=

{

c1

(n1

D1+

n3

D3

)

+ c2

(n2

D2− n3

D3

)}

≡ c2A1234 + c1A1243 , (4.8)



A1234 =

(1

D2+

1

D3

)

n2 −n1

D3, A1243 =

(1

D1+

1

D3

)

n1 −n2

D3, (4.9)


A1234 =

(1

D2+

1

D3− D1

(D1+D3)D3

)

n2 −D1

D1+D3A1243 . (4.10)


D1 +D2 +D3 = (s13 −m2) + (s14 −m2) + s12 = 0 , (4.11)


(s14 −m2)A1234 = (s13 −m2)A1243 . (4.12)



Atree4,1 =

(

T a3ı2j

T a4 i1

+ T a4ı2j

T a3 i1

s14 −m2

s13 −m2

)

A1234 . (4.13)







c1 − c2 = −c5 ⇔ n1 − n2 = −n5 , (4.14a)

c3 − c4 = c5 ⇔ n3 − n4 = n5 . (4.14b)




– 21 –

à BCJ amplitude rel.

Amplitude relations & basis

duality were shown to exist to all multiplicities [80, 92, 93], thus effectively proving the

duality at tree level for k ≤ 1.

By explicit calculations we have checked that the duality works for any quark-gluon

configurations up to eight particles. While we do not provide a proof for n > 8, k > 1,

we will show that the duality imposes the BCJ relations in QCD that are constitute a

well-defined subset of the pure-gluon BCJ relations.

In the pure-gluon case, the number of independent BCJ relations are (n−2)!− (n−3)!

and they are given in ref. [31]. A subset of those are linear in momentum invariants,

n−1∑

i=2

( i∑

j=2

sjn)

A(1, 2, . . . i, n, i+ 1, . . . , n − 1) = 0 , (4.27)

and in ref. [85] it was shown that relabelings of these simple equations could be used to

derive the more complicated relations involving higher powers of momentum invariants.

We found that the corresponding quark-gluon BCJ relations, for k massive quark lines,

are given by the general formula

n−1∑

i=2

( i∑

j=2

sjn −m2j

)

A(1, 2, . . . i, n, i + 1, . . . , n− 1) = 0 . (4.28)

where particle n is strictly a gluon, while the remaining (n − 1) particles can be of any

type: quark/antiquark/gluon. In the next section we will generalize this formula to include

the relations with higher powers of momentum invariants, as well as derive the number of

linearly independent BCJ relations, counted for n ≤ 8 in table 3.

In section 4.2 we derived a n = 5, k = 2 amplitude relation (4.18) with the permuted

leg n being a gluon. It has precisely the form (4.28). The four-point relation (4.12) from

section 4.1 does not have this precise form right away, it can be easily rewritten that way,

either as a sum over different insertions of the gluon leg 4,

(s24 −m2)A1243 + (s24 + s34 −m2)A1234 = 0 , (4.29)

or, equivalently, as a sum over insertions of the gluon leg 3,

(s23 −m2)A1234 + (s23 + s34 −m2)A1243 = 0 . (4.30)

For the pure-quark six-point amplitude in section 4.3, we note that eq. (4.28) is consis-

tent with the fact it had no BCJ amplitude relations despite obeying the color-kinematics

duality. Indeed, by definition pure-quark amplitudes, n = 2k, have no external gluons, and

thus eq. (4.28) gives no relations for them.

We derived the quark-gluon BCJ relations (4.28) as follows. We start with the primitive

amplitudes in the Melia basis (3.19),

{

A(1, 2,σ) =∑


± ni

Di

∣∣∣ σ ∈ Melia basis

}

, (4.31)

– 25 –

BCJ relations for pure-gluon amplitudes:

duality were shown to exist to all multiplicities [80, 92, 93], thus effectively proving the

duality at tree level for k ≤ 1.

By explicit calculations we have checked that the duality works for any quark-gluon

configurations up to eight particles. While we do not provide a proof for n > 8, k > 1,

we will show that the duality imposes the BCJ relations in QCD that are constitute a

well-defined subset of the pure-gluon BCJ relations.

In the pure-gluon case, the number of independent BCJ relations are (n−2)!− (n−3)!

and they are given in ref. [31]. A subset of those are linear in momentum invariants,

n−1∑

i=2

( i∑

j=2

sjn)

A(1, 2, . . . i, n, i+ 1, . . . , n − 1) = 0 , (4.27)

and in ref. [85] it was shown that relabelings of these simple equations could be used to

derive the more complicated relations involving higher powers of momentum invariants.

We found that the corresponding quark-gluon BCJ relations, for k massive quark lines,

are given by the general formula

n−1∑

i=2

( i∑

j=2

sjn −m2j

)

A(1, 2, . . . i, n, i + 1, . . . , n− 1) = 0 . (4.28)

where particle n is strictly a gluon, while the remaining (n − 1) particles can be of any

type: quark/antiquark/gluon. In the next section we will generalize this formula to include

the relations with higher powers of momentum invariants, as well as derive the number of

linearly independent BCJ relations, counted for n ≤ 8 in table 3.

In section 4.2 we derived a n = 5, k = 2 amplitude relation (4.18) with the permuted

leg n being a gluon. It has precisely the form (4.28). The four-point relation (4.12) from

section 4.1 does not have this precise form right away, it can be easily rewritten that way,

either as a sum over different insertions of the gluon leg 4,

(s24 −m2)A1243 + (s24 + s34 −m2)A1234 = 0 , (4.29)

or, equivalently, as a sum over insertions of the gluon leg 3,

(s23 −m2)A1234 + (s23 + s34 −m2)A1243 = 0 . (4.30)

For the pure-quark six-point amplitude in section 4.3, we note that eq. (4.28) is consis-

tent with the fact it had no BCJ amplitude relations despite obeying the color-kinematics

duality. Indeed, by definition pure-quark amplitudes, n = 2k, have no external gluons, and

thus eq. (4.28) gives no relations for them.

We derived the quark-gluon BCJ relations (4.28) as follows. We start with the primitive

amplitudes in the Melia basis (3.19),

{

A(1, 2,σ) =∑


± ni

Di

∣∣∣ σ ∈ Melia basis

}

, (4.31)

– 25 –

BCJ relations for quark-gluon QCD amplitudes: HJ, Ochirov

Bern, Carrasco, HJ

gluon ! proof by: de la Cruz, Kniss, Weinzierl

k \ n 3 4 5 6 7 8

0 1 1 2 6 24 120

1 1 1 2 6 24 120

2 - 1 2 6 24 120

3 - - - 4 16 80

4 - - - - - 30

β(n, k) =

{

(n− 3)! for k = 0, 1

(n− 3)!(2k − 2)/k! for 2 < 2k ≤ n

Table 4: Number of independent primitive amplitudes, β(n, k), in the full n-point amplitude withk distinguishable quark pairs and (n− 2k) gluons, after imposing the BCJ relations.

quark pair configurations. The (n − 2k) gluons are then free to be assigned to the slots

in-between the particles, except for the space inside the fixed sequence 1, 2, q. For each

gluon inserted the available number of slots increases, ranging from (2k − 2) to (n − 3).

Thus the length of the basis is

β(n, k) =


(2k − 2)!

k!(k − 1)!×(k − 1)!

︸︷︷︸


× (2k − 2)(2k − 1) . . . (n − 3)︸︷︷︸


=(n − 3)!(2k − 2)

k!. (4.34)

For the amplitudes that only have one quark/antiquark pair, k = 1, it is not possible

to pick the third particle to be a quark. So instead we pick it to be the gluon 3, giving

the basis A(1, 2, 3,σ). Except for the bars on the labels, this is the same basis as for the

pure-gluon case [31], thus the basis for k = 0, 1 is of size (n − 3)!. The basis counts for

different quark-gluon configurations are exemplified and summarized in table 4.

The full solution to the BCJ relations are given by first solving the numerators ni in

terms of the primitives in the BCJ basis (4.33), and then plugging them into the primitives

that are not part of this basis. Since the color-algebra basis (3.19) already has legs 1 and 2

next to each other, we only need to give the reduction formula for the primitives with the

first quark leg q separated from leg 2 by a set of gluonic legs α and followed by a mixed

set of quark-gluon legs β:

A(1, 2,α, q,β) . (4.35)

To simplify the subsequent formulas, we choose q = 3 and take the leg labels in sets α and

β to be consecutive numbers:

α ≡ {4, 5, . . . , p − 1, p} , q ≡ 3 , β ≡ {p + 1, p + 2, . . . , n− 1, n} . (4.36)

As already mentioned, α consists strictly of gluon legs, and the particles in β can be of

any type: quark/antiquark/gluon. The consecutive labeling choice can always be undone

in the final expressions by relabeling of legs 3, 4, . . . , n.

– 27 –

Basis:

k \ n 3 4 5 6 7 8

0 1 1 2 6 24 120

1 1 1 2 6 24 120

2 - 1 2 6 24 120

3 - - - 4 16 80

4 - - - - - 30

β(n, k) =

{

(n− 3)! for k = 0, 1

(n− 3)!(2k − 2)/k! for 2 < 2k ≤ n

Table 4: Number of independent primitive amplitudes, β(n, k), in the full n-point amplitude withk distinguishable quark pairs and (n− 2k) gluons, after imposing the BCJ relations.

quark pair configurations. The (n − 2k) gluons are then free to be assigned to the slots

in-between the particles, except for the space inside the fixed sequence 1, 2, q. For each

gluon inserted the available number of slots increases, ranging from (2k − 2) to (n − 3).

Thus the length of the basis is

β(n, k) =


(2k − 2)!

k!(k − 1)!×(k − 1)!

︸︷︷︸


× (2k − 2)(2k − 1) . . . (n − 3)︸︷︷︸


=(n − 3)!(2k − 2)

k!. (4.34)

For the amplitudes that only have one quark/antiquark pair, k = 1, it is not possible

to pick the third particle to be a quark. So instead we pick it to be the gluon 3, giving

the basis A(1, 2, 3,σ). Except for the bars on the labels, this is the same basis as for the

pure-gluon case [31], thus the basis for k = 0, 1 is of size (n − 3)!. The basis counts for

different quark-gluon configurations are exemplified and summarized in table 4.

The full solution to the BCJ relations are given by first solving the numerators ni in

terms of the primitives in the BCJ basis (4.33), and then plugging them into the primitives

that are not part of this basis. Since the color-algebra basis (3.19) already has legs 1 and 2

next to each other, we only need to give the reduction formula for the primitives with the

first quark leg q separated from leg 2 by a set of gluonic legs α and followed by a mixed

set of quark-gluon legs β:

A(1, 2,α, q,β) . (4.35)

To simplify the subsequent formulas, we choose q = 3 and take the leg labels in sets α and

β to be consecutive numbers:

α ≡ {4, 5, . . . , p − 1, p} , q ≡ 3 , β ≡ {p + 1, p + 2, . . . , n− 1, n} . (4.36)

As already mentioned, α consists strictly of gluon legs, and the particles in β can be of

any type: quark/antiquark/gluon. The consecutive labeling choice can always be undone

in the final expressions by relabeling of legs 3, 4, . . . , n.

– 27 –

Simple 1-loop examples

One-loop calculations

kinematic algebra:

diagrams:

ntri

(1, 2, 3, 4, `) = nbox

([1, 2], 3, 4, `) ,

nbub

(1, 2, 3, 4, `) = nbox

([1, 2], [3, 4], `) ,

nsnail

(1, 2, 3, 4, `) = nbox

([[1, 2], 3], 4, `) ,

ntadpole

(1, 2, 3, 4, `) = nbox

([[1, 2], [3, 4]], `) ,

nxtadpole

(1, 2, 3, 4, `) = nbox

([[[1, 2], 3], 4], `) .

vanish after integration

Ansatz for the box numerator: N =0,1,2 SQCD

ansatz for 4pt MHV amplitude with internal matter, in any SYM theory: HJ, Ochirov

diagrams:

M (N) =n

NY

i=1

mi

�

�

�

mi 2 {s, t, ` · kj , `2, µ2}o

momentum monomials:

ij =[1 2] [3 4]

h1 2i h3 4i�(2N )(Q) hi ji4�N ✓i✓jstate dependence:

VN = VN + V N ✓( vector multiplet: )

N = 4�N ß SUSY power-counting factor:

Unitarity cuts

−+

1−

2−3+

4+

(a)

−+

1−2−

3+4+

(b)

−+

1−

2−

3+

4+

(c)

Figure 14: Non-planar single-line cuts used to compute the numerators and amplitudes.

invariance, we can alternatively regard this as a flip of the chirality of the external vector

particles while keeping the internal matter unaltered. For consistent interactions, we also

need to flip the sign of parity-odd momentum invariants. The conjugated one-loop four-

point MHV numerators are then

ni(1, 2, 3, 4, !) = ni(1, 2, 3, 4, !)∣∣κij→κ!

ij , ε(1,2,3,#)→−ε(1,2,3,#) , (4.17)

where κ!ij = κkl marks the pair of legs {k, l} = {1, 2, 3, 4} \ {i, j} unmarked by κij . Com-

bining eqs. (4.17) and (4.16) give the desired extra constraint.

The ansatz (4.10) is ultimately constrained by the unitarity cuts. We choose to work

with the non-planar single-line cuts shown in fig. 14. These cuts can be constructed by

taking color-ordered six-point tree-level amplitudes and identifying a conjugate pair of

fundamental particles on opposite-site external legs. Because the tree amplitude is color

ordered, the identification of momenta on opposite ends of the amplitude does not produce

singularities corresponding to soft or collinear poles.

As the internal loop momenta are subject to only one constraint, !2 = 0, the single-

line cuts are sensitive to most terms in the integrand of the amplitude. Undetected terms

correspond to tadpoles and snails (external bubbles), which invariably integrates to zero

in a massless theory. Indeed, the tadpole and snail graphs previously described do not

contribute to this cut, instead these will be indirectly constrained through the kinematic

relations of the numerators.

Even after the symmetries and unitarity cuts are imposed on the numerators, there

remain free parameters in ansatz that can be interpreted as the residual generalized gauge

freedom of the current amplitude representation. For simplicity, we will fix this freedom

by suitable aesthetic means, as discussed in the next section.

4.4 The amplitude assembly

In this section, we provide the precise details on how to assemble the full one-loop gauge-

theory amplitude from the numerators and the color factors. This discussion will be valid

for all of the YM theories and amplitudes discussed in the subsequent sections.

The complete MHV (super-)amplitude can be written as

A1-loop4 =

∑

S4

(18Ibox +

1

4Itri +

1

16Ibub

), (4.18)

– 31 –

Parameters in ansatz fixed by unitarity cuts (unitarity method) Bern, Dixon, Dunbar, Kosower

N=1 SQCD: HJ, Ochirov

N=2 SQCD: Carrasco, Chiodaroli, Gunaydin, Roiban; Nohle; Ochirov, Tourkine, HJ, Ochirov

QCD: HJ, Ochirov YM + scalar: Nohle; HJ, Ochirov

We choose to fix the remaining four free parameters by imposing aesthetic or practical

constraints to obtain compact and manageable numerator expressions. Requiring that

the snail numerator nsnail vanishes for any on-shell external momenta gives two additional

conditions. Finally, requiring that the parity-even part of the s-channel triangle defined in

eq. (4.2) is proportional to s gives two more conditions. The latter implicitly enforces no

dependence on κ12 and κ34 in that triangle.9

Having thus solved for all free parameters, we obtain the following box numerator:

nN=2,fundbox =(κ12 + κ34)

(s− "s)2

2s2+ (κ23 + κ14)

"2t2t2

+ (κ13 + κ24)st+ (s+ "u)2

2u2

− 2iε(1, 2, 3, ")κ13 − κ24

u2+ µ2

(κ12 + κ34s

+κ23 + κ14

t+κ13 + κ24

u

),

(4.28)

where the short-hand notation (4.5) for loop-momentum invariants is used and the param-

eters κij encoding the external multiplets are defined in eq. (4.8).

The other numerators are given by the kinematic relations in eqs. (4.2) and (4.3). To

be explicit, the triangle numerator is

nN=2,fundtri =(κ23 + κ14)

s(t− 2"t)

2t2− (κ13 + κ24)

s(u− 2"u)

2u2

+ 2iε(1, 2, 3, ")κ23 − κ14

t2+ 2iε(1, 2, 3, ")

κ13 − κ24u2

,(4.29)

and the internal bubble numerator is

nN=2,fundbub = s

(κ23 + κ14t

− κ13 + κ24u

). (4.30)

In principle, we could ignore any further contributions to the integrand of the am-

plitude, since snail (external bubble) and tadpole diagrams should integrate to zero in

dimensional regularization. However, for completeness, we give the result for the only

non-vanishing diagram of this type – the snail. Because of N = 2 supersymmetry power

counting, it must include an overall factor k24 (in analogy to a one-loop propagator correc-

tion), which vanishes on shell. This contribution is not visible in our ansatz, because the

external legs were placed on shell from the very start. Nevertheless, a careful analysis of a

singular two-particle unitarity cut reveals that the numerator of the snail diagram shown

in fig. 11 is given by

nN=2,fundsnail = −k24

2

(κ23 + κ14t

− κ13 + κ24u

). (4.31)

As the snail graph has a propagator 1/k24 , the above numerator gives a finite contribution

to the integrand, after the 0/0 is properly canceled out. Once this is done the snail-diagram

contribution can be included into the amplitude (4.18) as

1

4

∑

S4

Isnail , (4.32)

9Alternatively, one can demand that the bubble numerator vanishes, as was done in refs. [10, 28, 57].

This also fixes the last two parameters, but gives a more complicated expression for the triangle.

– 34 –

where we define

nN=1,eveni ≡ nN=1,fund

i + nN=1,fundi , (4.38a)

nN=1,oddi ≡ nN=1,fund

i − nN=1,fundi . (4.38b)

According to eq. (2.11), summing over the fundamental and antifundamental one-loop

numerators effectively corresponds to promoting the N = 1 multiplet to the adjoint repre-

sentation, which by eq. (2.43) is equivalent to a contribution of a N = 2 multiplet. Thus

we have the following equalities:

nN=1,eveni = nN=2,adj

i = nN=2,fundi . (4.39)

Since we have already found compact expressions for the N = 2 numerators in sec-

tion 4.5, we can now focus entirely on the parity-odd N = 1 numerators. Unlike the

parity-even ones, they have the expected amount of supersymmetry, hence Neff = 1. Using

the ansatz (4.10) as before, except with N = 4−Neff = 3, gives us 546 free parameters to

solve for. Similarly to the procedure in the previous section, we can immediately eliminate

half of the parameters by imposing the defining property that the box numerator should

be odd under matter conjugation:

noddbox = −nodd

box , (4.40)

which reduces the problem to 273 undetermined parameters.

Next comes the dihedral symmetry of section 4.3: the cyclic symmetry (4.15) fixes 208

parameters. An additional 20 are constrained by the flip relation (4.4), which becomes a

symmetry after imposing eq. (4.40).

Out of the remaining 45 free parameters, 20 are fixed by the four-dimensional unitar-

ity cuts shown in fig. 14. Requiring that the snail numerator vanishes on shell gives 17

additional constraints. Demanding that the power counting of the numerator is at worst

!m for one-loop m-gons gives 3 more conditions, leaving only five parameters to fix. One

can check that four of them correspond to the genuine freedom of the color-kinematics

representations, and one parameter is determined by D-dimensional unitarity cuts. How-

ever, since it is difficult to analytically continue four-dimensional chiral fermions to D > 4,

computing the correct unitarity cut is challenging. Instead, in our final representation, we

denote this unfixed parameter by a and manually choose the remaining four parameters to

the values that give more compact expressions.

The box numerator for the N = 1 odd contribution is then given by

nN=1,oddbox =(κ12 − κ34)

(!s − s)3

2s3+ (κ23 − κ14)

!3t2t3

+ (κ13 − κ24)1

2

( !3uu3

+3s!2uu3

− 3s!uu2

+s

u

)

− 2iε(1, 2, 3, !)(κ13 + κ24)2!u − u

u3− aµ2(κ13 − κ24)

s− t

u2,

(4.41)

– 36 –

N=2 SQCD

where Isnail is defined by eqs. (4.19) or (4.22) with the respective color factors

csnail = Tr([[T a1 , T a2 ], T a3 ]T a4) or cadjsnail = fa1a2cf ca3df bdef ea4b . (4.33)

Of course, the snail diagram (4.32) still integrates to zero, so it would be justified to

drop it. Nevertheless, we choose to explicitly display the snail graph because there is some

potential interest in the planar N = 2 integrand, in analogy with the recent advances with

the planar N = 4 integrand [63, 64]. Lastly, the two remaining numerators ntadpole and

nxtadpole are manifestly zero, consistent with naive expectations in N = 2 supersymmetric

theories.

If we now assemble the full amplitude (4.18) and recast it into the color-ordered

form (4.21), we obtain the two inequivalent partial amplitudes in D = 4 − 2ε, which

are most easily expressed as

AN=2,fund4 (1−, 2−, 3+, 4+) =

i〈12〉2[34]2

(4π)D/2

{− 1

stI2(t)

}, (4.34a)

AN=2,fund4 (1−, 2+, 3−, 4+) =

i〈13〉2[24]2

(4π)D/2

{− rΓ2u2

(ln2

(−s

−t

)+ π2

)+

1

suI2(s) +

1

tuI2(t)

},

(4.34b)

where I2 is the standard scalar bubble integral

I2(t) =rΓ

ε(1− 2ε)(−t)−ε . (4.35)

Here and below, the integrated expressions are shown up to O(ε), and the standard pref-

actor of dimensional regularization is

rΓ =Γ(1 + ε)Γ2(1− ε)

Γ(1− 2ε). (4.36)

Due to eq. (4.26) the partial amplitudes (4.34) are the coefficients of both the clockwise

and counterclockwise fundamental color traces in the color-dressed amplitude. Moreover,

they coincide with the primitive amplitudes A1-loop,adj4 with adjoint N = 1 matter in the

loop (same as adjoint N = 2 in our convention, see eq. (2.43)), which is the version that is

best known in the literature [56, 65].

4.6 The amplitude with N = 1 fundamental matter

In this section, we work out the numerators of the four-point one-loop amplitude with a

N = 1 fundamental matter multiplet ΦN=1 circulating in the loop and adjoint vectors

VN=1 on the external legs. The result is the first known color-kinematics representation of

this amplitude.

At one loop, the amplitude, along with its numerators, can be naturally decomposed

into two simpler components: the parity-even and parity-odd contributions:

nN=1,fundi =

1

2nN=1,eveni +

1

2nN=1,oddi , (4.37a)

nN=1,fundi =

1

2nN=1,eveni − 1

2nN=1,oddi , (4.37b)

– 35 –

N=2 SQCD integrated amplitude:�








theories.




AN=2,fund4 (1−, 2−, 3+, 4+) =

i〈12〉2[34]2

(4π)D/2

{− 1

stI2(t)

}, (4.34a)

AN=2,fund4 (1−, 2+, 3−, 4+) =

i〈13〉2[24]2

(4π)D/2

{− rΓ2u2

(ln2

(−s

−t

)+ π2

)+

1

suI2(s) +

1

tuI2(t)

},

(4.34b)


I2(t) =rΓ

ε(1− 2ε)(−t)−ε . (4.35)



rΓ =Γ(1 + ε)Γ2(1− ε)

Γ(1− 2ε). (4.36)










this amplitude.



nN=1,fundi =

1

2nN=1,eveni +

1

2nN=1,oddi , (4.37a)

nN=1,fundi =

1

2nN=1,eveni − 1

2nN=1,oddi , (4.37b)

– 35 –








theories.




AN=2,fund4 (1−, 2−, 3+, 4+) =

i〈12〉2[34]2

(4π)D/2

{− 1

stI2(t)

}, (4.34a)

AN=2,fund4 (1−, 2+, 3−, 4+) =

i〈13〉2[24]2

(4π)D/2

{− rΓ2u2

(ln2

(−s

−t

)+ π2

)+

1

suI2(s) +

1

tuI2(t)

},

(4.34b)


I2(t) =rΓ

ε(1− 2ε)(−t)−ε . (4.35)



rΓ =Γ(1 + ε)Γ2(1− ε)

Γ(1− 2ε). (4.36)










this amplitude.



nN=1,fundi =

1

2nN=1,eveni +

1

2nN=1,oddi , (4.37a)

nN=1,fundi =

1

2nN=1,eveni − 1

2nN=1,oddi , (4.37b)

– 35 –

Bern, Dixon, Dunbar, Kosower; Bern, Morgan

Using the QCD numerators to get GR Pure Einstein gravity can be obtained from the QCD numerators:

M(1)4 =

X

S4

X

i={B,t,b}

ZdD`

(2⇡)D1

Si

nVi ni

V 0 � nmi ni

m0 � nmi ni

m0

Di

The YM square contains dilaton & axion, which has to be subtracted out

…and similarly for triangle and bubble

quark quark

HJ, Ochirov

Gives correct pure GR amplitude (cf. Dunbar & Norridge)

Loop-level application QCD

Two-loop 5pt all-plus-helicity amplitude in pure YM computed to all orders in Nc using the DDM basis and BCJ relations: Badger, Mogull, Ochirov, O’Connell (arXiv:1507.08797)

Summary !   Color-kinematics duality implies kinematic Lie algebra relations satisfied by the numerators of gauge theory amplitudes

!   Generalized color-kinematics duality to QCD tree amplitudes

!   New color decomposition of QCD tree amplitudes

!   BCJ amplitude relations between primitives of QCD !   Checks: Explicitly up to 8pts tree level, proof color decompositon (Melia)

proof BCJ relations (Weinzierl, et al.) !   Constructed one-loop 4pt amplitude in N=1,2 SQCD and QCD such that the

duality is manifest.

!   Useful for construction of QCD loop amplitudes as well as pure Einstein gravity amplitudes

c Color-Kinematics Duality for QCD Amplitudes · 2020. 11. 19. · 3 Henrik Johansson Uppsala U. &...

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